# Gibbs-Duhem Relation/Entropic Equations of State

1. Nov 1, 2013

### mateomy

I'm working through Callen (ch. 3.4) right now and I'm trying to follow his explanation of the Gibbs-Duhem relation:

Beginning with the ideal gas characterizations
$$PV = nRT$$
$$U = cnRT$$
there is some rearranging of terms and we are shown the relations
$$\frac{P}{T}=\frac{nR}{V}$$
$$\frac{1}{T}=\frac{cNR}{U}$$
and recognizing the further definitions of $n/V \equiv v$ and $N/U \equiv u$.
Then Callen states 'from these two entropic equations of state we find the third equation of state'
$$\frac{\mu}{T} = \,function\, of\, u,\, v \,\,\,\,\,\,[1]$$
by integrating the Gibbs-Duhem relation
$$d\left(\frac{\mu}{T}\right) = ud\left(\frac{1}{T}\right) + vd\left(\frac{P}{T}\right)$$

I'm not seeing where he gets [1] from. Is that just a definition? I noticed a few sections before this he defined:
$$d\mu = -sdT + vdP$$
I'm missing something but I don't know what. Just looking for some clarification, thanks.

Last edited: Nov 1, 2013
2. Nov 2, 2013

### dextercioby

He says that the integrated version [1] comes from the differential one called Gibbs-Duhem. So your question is actually where Gibbs-Duhem comes from and that should be answered by Callen.